Integral Apollonian circle packings and prime curvatures

TL;DR

This paper demonstrates that primitive integral Apollonian circle packings contain infinitely many prime curvatures by employing the circle method and binary quadratic forms.

Contribution

It introduces a novel approach using the circle method and quadratic forms to show prime curvatures appear in Apollonian packings.

Findings

Primitive integral packings contain infinitely many prime curvatures

Application of the circle method to geometric number theory

Connection between circle packings and prime number distribution

Abstract

It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary quadratic forms.